The chain on my internally geared hub bicycle is now too long even when the rear wheel is entirely pulled back. How do I tell whether it's sufficient to shorten the chain by removing some links, or if I need to buy a new chain altogether?

If it was an externally geared bike I would say replace it. Given that it's an internally geared bike and therefore basically a singlespeed in terms of what the chain touches, I would be less inclined to change it. You can run a chain on a single speed fr a very, very long time without issues. The components will all wear together and yes you will likely have to replace the chainring, cog, and chain all at the same time but you can get away with lots more miles on a single setup without any adverse effects compared to an externally geared configuration

A chain with this many miles on it almost certainly has a large amount of what's called "chain stretch". As chains wear, the distance in between the link gaps increases. (This page on chain maintenance will tell you more than you need to know on the subject.) When this gets past a certain point, the chain needs to be replaced. If this isn't done, increased wear to the drivetrain sprockets almost certainly will result, since the chain no longer fits over the sprockets properly. (Since this is bike has an IGH, this problem will still exist but to a lesser degree, due to the lack of derailer gears.)

I highly recommend you not only replace your chain but also have the sprocket and chainring examined; they may need to be replaced as well. Once this is done, you'll notice the bike shifts more smoothly and works better in general.